Advances in Pure Mathematics, 2012, 2, 1-5
http://dx.doi.org/10.4236/apm.2012.21001 Published Online January 2012 (http://www.SciRP.org/journal/apm)
Real Hypersurfaces in 2 
P
and 2 
H
Equipped with
Structure Jacobi Operator Satisfying =llL


n

Konstantina Panagiotidou, Philippos J. Xenos
Mathematics Division, School of Technology, Aristotle University of Thessaloniki, Thessaloniki, Greece
Received June 27, 2011; revised August 4, 2011; accepted August 20, 2011
ABSTRACT
P
Recently in [1], Perez and Santos classified real hypersurfaces in complex projective space 3n

for , whose Lie
derivative of structure Jacobi operator in the direction of the structure vector field coincides with the covariant deriva-
tive of it in the same direction. The present paper completes the investigation of this problem studying the case n = 2 in
both complex projective and hyperbolic spaces.
Keywords: Real Hypersurfaces; Complex Projective Space; Complex Hyperbolic Space; Lie Derivative; Structure
Jacobi Operator
1. Introduction
A complex n-dimensional Kaehler manifold of constant
holomorphic sectional curvature c is called a complex
space form, which is denoted by n
M
c
n
. A complete
and simply connected complex space form is complex
analytically isometric to a complex projective space
a complex Euclidean space n or a complex hy-
perbolic space
P
n
H
 if , or c respec-
tively.
>0c=0 <0
n
c
The study of real hypersurfaces was initiated by Ta-
kagi (see [2]), who classified homogeneous real hyper-
surfaces in
P

n
and showed that they could be divided
into six types, which are said to be of type A1, A2, B, C, D
and E. Berndt (see [3]) classified homogeneous real hy-
persurfaces in
H

n
P
n
with constant principal curvatures.
Okumura (see [4]) in and Montiel and Romero
(see [5]) in
H
 gave the classification of real hyper-
surfaces satisfying relation =0AA

 .
Ki and Liu (see [6]) have given the above classifica-
tion as follows:
Theorem A Let M be a real hypersurface of


n
M
c
0cA
,
, (n ≥ 2). If it satisfies=0A

 
n
, then M is lo-
cally congruent to one of the following hypersurfaces:
 In case
P



1
A
a geodesic hypersphere of radius r, where
π
0<< 2 r

,

2
A
a tube of radius r over a totally geodesic k
P
,

2kn1 , where π 0< <.
2
r
n
H

 In case


n
0
A
a horosphere in
H
, i.e. a Montiel tube,


1
A
a geodesic hypersphere or a tube over a hyper-
plane 1n

H
,


k
2
A
a tube over a totally geodesic
H



12kn



n
.
M
c0c , (Let M be a real hypersurface in

).
Then an almost contact metric structure (,,, )
g

 can
be defined on M. The structure vector field

is called
principal if =A


holds on M, where A is the shape
operator of M in


n
M
c and

is a smooth function.
A real hypersurface is said to be a Hopf hypersurface if

is principal.
The Jacobi operator field with respect to X on M is de-
fined by


,RXX
=X
, where R is the Riemmanian curva-
ture of M. For

the Jacobi operator is called
structure Jacobi operator and is denoted by


=,lR

 .
It has a fundamental role in almost contact manifolds.
Many differential geometers have studied real hypersur-
faces in terms of the structure Jacobi operator.
The Lie derivative of the structure Jacobi operator
with respect to

was investigated by Perez, Santos,
Suh (see [7]). More precisely, they classified real hyper-
surfaces in n
P
3
=0l
2
(n), whose structure Jacobi op-
erator satisfies the condition:

L. Ivey and Ryan
(see [8]) classified real hypersurfaces satisfying the same
condition in
P
2
and
H
.
The study of real hypersurfaces whose structure Jacobi
operator is parallel is a problem of great importance. In
[9] the nonexistence of real hypersurfaces in nonflat
C
opyright © 2012 SciRes. APM

K. PANAGIOTIDOU ET AL.
2





,= ,,
,= ,
complex space form with parallel structure Jacobi opera-
tor () was proved. In [10] a weaker condition
(-parallelness), X for any vector field X or-
thogonal to
g
XY gXYXY
gX YgXY
 



 
=,
=,
X
X
AX
YYAXgAXY
=0l
=0l

, was studied and it was proved the non-
existence of such hypersurfaces in case of n
P
3
n
(n).
The parallelness of structure Jacobi operator in combina-
tion with other conditions was another problem that was
studied by many others such as Ki, Kim, Perez, Santos,
Suh (see [11,12]).
Recently Perez-Santos (see [1]) studied real hypersur-
faces in
P
 for , whose structure Jacobi opera-
tor satisfies the relation:
3n
=.ll

L
P H
P
H
(1)
In the present paper we go on studying the same prob-
lem for 2  and 2. We prove the following theo-
rem:
Main Theorem Let M be a real hypersurface in 2
or 2, whose structure Jacobi operator satisfies rela-
tion (1). Then M is locally congruent to: a geodesic
sphere of radius r, where π <2 r0< with π
4
r, or to
a tube of radius π =4 r over a holomorphic curve in
2 and to a horosphere, a geodesic sphere or a tube
over
P
1
H

A
in 2i or to a Hopf hypersurface in 2
with
HH
=0

.
2. Preliminaries
Let M be a connected real hypersurface immersed in a
nonflat complex space form



,
n
M
cG 0c
=
, (), with
almost complex structure J of constant holomorphic sec-
tional curvature c. Let N be a unit normal vector field on
M and
J
N

. For a vector field X tangent to M we
can write
 
=

J
XX  XN , where φX and


X
N

are the tangential and the normal component of JX re-
spectively. The Riemannian connection

in


n
M
c
and in M are related for any vector fields X, Y on M: 


=,
YY
X
XgAYXN
=
XNAX

where g is the Riemannian metric on M induced from G
of

n
M
c and A is the shape operator of M in


n
M
c
(,,)
.
M has an almost contact metric structure



in-
duced from J on

n
M
c where

is a (1,1) tensor field
and

is a 1-form on M such that

,= ,,
g
XY  G JXY
 
= ,

=,
X
gX  GJXN
 
=0,X 
(see [13]). Then we have


2=,
=0, =1
XX X 
 
 (2)
(3)




 

(4)
Since the ambient space is of constant holomorphic
sectional curvature c, the equations of Gauss and Codazzi
for any vector fields X, Y, Z on M are respectively given
by
 
 
 
,= ,,,
4
,2,
,,
c
RXYZgYZXgXZY gYZX
gXZYgXYZ
gAYZAXgAXZAY


 


 



(5)
 

=4
2,
XY
c
A
YAX XYYX
gXY


  



PM
(6)
where R denotes the Riemannian curvature tensor on M.
For every point

, the tangent space can
be decomposed as following:
P
TM

=
P
T Mspanker

 
where




=:=0kerXTMX  P. Due to the
above decomposition, the vector field
A

is decom-
posed as follows:
A
U=

   (7)
where =



and

1
=U ker





0
, pro-
vided that


.
All manifolds are assumed connected and all mani-
folds, vector fields etc are assumed smooth (C

).
3. Auxiliary Relations
2
P
2 or Suppose now that the ambient space is
H
,
(i.e.


2c0c,
M

), then we consider V be the open
subset of points P

M, such that there exists a neighbor-
hood of every P, where =0

and the open subset
of points Q of M such that there exists a neighborhood of
every Q, where

0a

. Since,

is a smooth function
on M, then is an open and dense subset of M. V
Proposition 3.1 Let M be a real hypersurface in


2
M
c, equipped with structure Jacobi operator satis-
fying (1). Then,

is principal on V.
A Proof: The relation (7) on V, takes the form


U
X
UYZ

. From (5) for


,

 we obtain:
.
4
c
lU U (8)

 

K. PANAGIOTIDOU ET AL. 3
Due to the definition of Lie derivative, the relation (1)
for X

 yields: l

0


0
. The latter, because of the
first relation of (4) and (8) implies


A
, hence
0

. Therefore,

is principal on V. □
On if =0

, then

is principal. In what fol-
lows we work on W (W), which is the open subset
of points such that

0 Q

 .
Lemma 3.2 Let M be a real hypersurface in


2
M
c.
Then the following relations hold on W:
2
,
44
cc
A
UU

AUU
 






 (9)
2
,
44
UU
cc
UU


, U


 

 

 (10)
12
,,
4
UU
c
UUU U  

 3 UU

 (11)
2
23
,
UU

14
,
U
U
c
UU
UU


 









3
,, 

,,UU


 
(12)
where are smooth functions on M.
12
Proof: If



=,=,
is an orthonormal basis, then
because of (7) we have:
A
UU UAUUU

   (13)
where

,

and

are smooth functions on M.
The first relation of (4), because of (13) yields:
=,
=.
U
U


= ,
U
UU UU





 
 (14)
The relation (5), using (13) can be written:
2
4
c
lU
lU U
,
.
4
U U
cU






 








.
lX X
l
(15)
By the definition of Lie derivative, the relation (1)
takes the form:


The latter for X 


,U


0lU implies lU = 0,


and then from (15) we obtain:
2
44
cc

  0, ,.


 

,
XX
(16)
The relations (13) and (14), because of (16) imply (9)
and (10) respectively.
From the well known relation:
,,
X
gYZgYZ
,,
gY Z
for
X
YZ


,,UU

 , using (16) we obtain (11) and
(12), where 1

, 2

and 3

are smooth functions on M.
□

,,UU   The Codazzi equation for X, Y , because
of Lemma 3.2 yields:
2
2
41,Uc 


 


(17)
22
3
1,
44
cc
 
 

 


(18)

2
2
4,Uc


  (19)
2
2
4,
c


 

(20)
3
3,
4
c
U 


 



(21)
2
2
1,
24
cc
U





 



(22)
22
13.
44
cc
U





 
 
 
 
(23)
The Riemannian curvature on M satisfies (5) and on
the other hand is given by the relation


,RXYZZZ Z

,,UU  


,
XY YXXY . From these two
relations and because of (16) for X, Y we
obtain:
2
3
21
2
22 3
12
242
c
cc
UU
c




 
 


 


(24)


2
312 3
4
c
U


 


 



(25)
3213 3
42
cc
U 



 


34
(26)
Relation (23), because of (18), (21) and (22), yields:

 (27)


and so relation (18) becomes:
2
2
14.
44
cc 








(28)
Differentiating the relations (27) and (28) with respect
to U and

respectively and substituting in (25) and
due to (19), (20) and (27) we obtain:


22
224 0c
 . (29)


K. PANAGIOTIDOU ET AL.
4
Owing to (29), we consider , (W) the open
subset of points , where in a neighbor-
hood of every Q.
1
W1W
20
 QW
Lemma 3.3 Let M be a real hypersurface in


2
M
c
1
W
2
c
,
equipped with Jacobi operator satisfying (1). Then
is empty.
Proof: Due to (29) we obtain: on 1 W.
Differentiation of the last relation along
2
24  

and taking
into account (19), (20) and yields: ,
which is a contradiction. Therefore, is empty. □
2
24
2
c  =0c
2=0

0.
1
On W, because of Lemma 3.3, we have , hence
the relations (17), (19) and (20) become:
W
UU

   
Using the last relations and Lemma 3.2 we obtain:


,0,U   UU   





.cU
22
,
1416
4
U
UU

 

 


0cU  
WWQW

 

Combining the last two relations we have:

22
416   . (30)
Let 2, (2) be the set of points W

, for
which there exists a neighborhood of every Q such that
. So from (30) we have: 16 .
Differentiating the last relation with respect to

U  0 22 4c  
U

and
taking into account (21), (22), (27) and (28), we have:
, which is impossible. So 2 is empty. Hence,
on W we have . Then, relations (21) and
(28), because of (27) imply respectively:
20
 W

U  0
2
4c


2
and
2
5
1

   . Relation (26), because of

0U 

,
2
4c

 and (27) yields: 1  2


. Substitution of 1

in 1
2
52

  
yields: 322

 . Differentiation of
the last one along U

and taking into account (22) leads
to: 0

, which is a contradiction. So we obtain the
following proposition:
Proposition 3.4 Let M be a real hypersurface in
2

M
c, equipped with Jacobi operator satisfying (1).
Then M is a Hopf hypersurface.
4. Proof of Main Theorem
Since M is a Hopf hypersurface, we can write
A
ZZ


and
A
ZZ

  with


,,
Z
Z

 , being a local or-
thonormal basis and the following relation holds:

24
c  


(Corollary 2.3 [14]). Furthermore,
due to Theorem 2.1 [14], we have that

is constant.
From (5), due to
A
ZZ

 and
A
ZZ

  we
get:
,.
44
cc
lZZl ZZ





 
 
 
 
(31)
The relation (1), because of (31) implies:


0  

, for
X
Z

and , for

0   
X
Z


. Combining the last two relations leads to:

20

2
P
, M is locally
0
. If

, in case of


congruent to a tube of radius π
4
2
over a holomorphic
P
curve in 
0A
due to [15] and to a Hopf hypersurface
with 2
H
. If


in 0

, the last relation im-
plies:



and we obtain:


0, ,
A
AX XTM  
which because of Theorem A completes the proof of
Main Theorem.
5. Acknowledgements
The first author is granted by Foundation Alexandros S.
Onasis. Grant Nr: G ZF 044/2009-2010. The authors
would like to thank Professors Juan de Dios Perez and
Patrick J. Ryan. Their answers improved the proof of
main theorem. Finally, the authors express their grati-
tude to the referee who has contributed to improve the
paper.
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